CM-points on straight lines

Let $\tau$ be an imaginary quadratic number with ${\mathrm{Im}\tau>0}$ and let $j$ denote the $j$-invariant function. According to the classical theory of Complex Multiplication, the complex number $j(\tau)$ is an algebraic integer. A CM-point in $\mathbb{C}^2$ is a point of the form $(j(\tau_1),j(\tau_2))$, where both $\tau_1$ and $\tau_2$ are imaginary quadratic numbers. In 1998 Yves André proved that a non-special (the notion will be defined during the talk) irreducible plane curve ${F(x_1,x_2)=0}$ may have only finitely many CM-points. This was the first non-trivial contribution to the celebrated André-Oort conjecture. Relying on recent ideas of Lars Kühne, we obtain a very explicit version of this result for straight lines defined over $\mathbb{Q}$: with ``obvious'' exceptions, a CM-point cannot belong to such a line. Kühne himself proved this for the line ${x_1+x_2=1}$. A joint work with Bill Allombert and Amalia Pizarro-Madariaga.